We invite the reader who is interested in a formal exposition of perturbed Markov chain theory to consult [1]} and [2]}, while in the following we simply make use of the resistance function \(r: X \times X \rightarrow \mathbb {R^+} \cup \lbrace \infty \rbrace \) , where \(r(x,x^{\prime })\)  indicates the minimal amount of perturbations required to move the system from \(x\)  to \(x^{\prime }\)  in one unit of time. If \(r(x,x^{\prime }) = 0\)  then the system moves from \(x\)  to \(x^{\prime }\)  with positive probability in the unperturbed dynamics, i.e., \(T_{xx^{\prime }}>0\) , while \(r(x,x^{\prime }) = \infty \)  is interpreted as impossibility of moving from \(x\)  to \(x^{\prime }\)  in one unit of time even when perturbations are allowed.
